2025 BCA 4th Semester Mathematics-II Question Paper | Purnea University

UniversityPurnea University

CourseBCA

SemesterIV

SubjectMathematics-II

Year2025

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CODE: BCA-M202

PURNEA UNIVERSITY, PURNEA

B.C.A. (Semester - IV) Examination, June-2025

Subject: Mathematics-II

TIME: 3 HOURS

FULL MARKS: 70

Answer any FIVE questions from total sheet

Prove that the statement $(P \rightarrow Q) \leftrightarrow (\neg Q \rightarrow \neg P)$ is a tautology using truth tables.

Briefly discuss propositional logic. Also define tautology, contradiction, and contingency with suitable examples.

(a) Translate the statement "Every student who studies hard will pass the exam" into Predicate logic.

(b) Construct a truth table for $(P \leftrightarrow Q) \leftrightarrow (\neg P \leftrightarrow \neg Q)$ and analyze the result.

(a) Prove that $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$.

(b) Prove De Morgan's laws for sets: $(A \cap B)' = A' \cup B'$ and $(A \cup B)' = A' \cap B'$.

(a) Define a graph. Differentiate between simple graph, multigraph, and pseudo-graph.

(b) What is a connected graph? How does it differ from a disconnected graph?

(a) What is an existential quantifier? Give an example.

(b) Define an equivalence relation with an example.

(a) Check if the vectors $(1,1)$ and $(2,2)$ are linearly independent.

(b) Find the determinant of the matrix: $\begin{bmatrix} 22 & -11 \\ -5 & -7 \end{bmatrix}$

(a) What is the difference between universal and existential quantifiers?

(b) Define a surjective mapping with an example.

(a) What is the role of truth tables in logic?

(b) What is a basis of a vector space? Provide an example.

10.

Simplify $\neg(P \rightarrow Q) \lor (Q \rightarrow P)$ using logical equivalences.

Detailed Solutions Outline

Step-by-Step Layouts

Q1 Solution (Tautology Proof):

To prove $(P \rightarrow Q) \leftrightarrow (\neg Q \rightarrow \neg P)$ is a tautology, construct a truth table with headers: $P, Q, \neg P, \neg Q, P \rightarrow Q, \neg Q \rightarrow \neg P$. The last column will consist entirely of $True$ (T).